Ricci solitons, Ricci flow, and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua

TL;DR

The work develops a boundary-aware framework for solving the Einstein-DeTurck equation via Ricci-DeTurck flow, proving a maximum principle that rules out nontrivial Ricci solitons under broad boundary conditions and showing that the flow preserves classes of geometries including asymptotically locally hyperbolic and extremal-horizon spacetimes. It demonstrates the existence and stability of a fixed point in a five-dimensional AdS geometry with Schwarzschild boundary conditions, corresponding to the gravity dual of a strongly coupled CFT on Schwarzschild in the Unruh or Boulware vacua, and extracts the leading large-N stress tensor, which remains regular on both horizons. The combination of analytic soliton-ruling principles and robust numerics yields a concrete Einstein metric solution accessible via a simple, stable flow, with implications for AdS/CFT and braneworld scenarios. The results show that, under suitable boundary data, Ricci-DeTurck flow is a reliable method to generate Einstein metrics and that the associated CFT stress tensor can remain well-behaved in nontrivial black hole backgrounds.

Abstract

The elliptic Einstein-DeTurck equation may be used to numerically find Einstein metrics on Riemannian manifolds. Static Lorentzian Einstein metrics are considered by analytically continuing to Euclidean time. Ricci-DeTurck flow is a constructive algorithm to solve this equation, and is simple to implement when the solution is a stable fixed point, the only complication being that Ricci solitons may exist which are not Einstein. Here we extend previous work to consider the Einstein-DeTurck equation for Riemannian manifolds with boundaries, and those that continue to static Lorentzian spacetimes which are asymptotically flat, Kaluza-Klein, locally AdS or have extremal horizons. Using a maximum principle we prove that Ricci solitons do not exist in these cases and so any solution is Einstein. We also argue that Ricci-DeTurck flow preserves these classes of manifolds. As an example we simulate Ricci-DeTurck flow for a manifold with asymptotics relevant for AdS_5/CFT_4. Our maximum principle dictates there are no soliton solutions, and we give strong numerical evidence that there exists a stable fixed point of the flow which continues to a smooth static Lorentzian Einstein metric. Our asymptotics are such that this describes the classical gravity dual relevant for the CFT on a Schwarzschild background in either the Unruh or Boulware vacua. It determines the leading O(N^2) part of the CFT stress tensor, which interestingly is regular on both the future and past Schwarzschild horizons.

Figures (7)

• Figure 1: Maximum value of $\frac{\partial T}{\partial \lambda}$ as a function of flow time $\lambda$ (black dots), and fit of the late time behaviour of this function to a power law $a\,\lambda^{-p-1}$ (red).
• Figure 2: Maximum value of $\phi$ in the whole domain as a function of the Ricci flow time $\lambda$ for different spatial resolutions obtained using the quasi-spectral code. The blue curve corresponds to the $20\times 20$ data, the red one to the $30\times 30$ data and the black curve to the $40\times 40$ data. The value of $\phi_{\textrm{max}}$ at the fixed point (constant section of the curves above) decreases as the spatial resolution is increased.
• Figure 3: Saturating value of $\phi_\textrm{max}$ as a function of the number of grid points $N$ (in either the $r$ or $x$ directions) for the quasi-spectral code. $\phi_\textrm{max}$ converges to zero exponentially with $N$, as expected for a suitably smooth solution.
• Figure 4: Embedding into hyperbolic space, $ds^2 = \frac{\ell^2}{z^2} \left( dz^2 + dR^2 + R^2 d\Omega_{(2)}^2 \right)$, of the spatial cross sections of the horizon along the flow as curves $R(z)$. The dashed line corresponds to the initial data, for which the horizon is round, and the thick black line is the embedding of the horizon of the fixed point. The snapshots are drawn at intervals of $\lambda$ of 0.05.
• Figure 5: Evolution of $(C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma})^{\frac{1}{4}}\ell$ along the flow. The snapshots correspond to $\lambda=0, 0.1,0.2, 0.5, 1$ and the fixed point. As shown in the plot, the Weyl tensor remains zero at the Poincare horizon $(r=1)$ and also at the AdS boundary $(x=1)$. The latter suggests that the ALH boundary conditions are preserved by the flow.